Abstract

We called graph G non-singular if adjacency matrix A (G) of G is non-singular. A connected graph with n vertices and n-1, n and n+1 edges are called the tree, the unicyclic graph and the bicyclic graph. Respectively, as we all know, each connected bicyclic graph must contain ∞(a,s,b) or θ(p,l,q) as the induced subgraph. In this paper, by using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained.

Highlights

  • This paper considers only finite undirected simple graphs

  • By using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained

  • Let G be a graph with order n, the matrix A is defined as the adjacency matrix of graph G, which is de

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Summary

Introduction

This paper considers only finite undirected simple graphs. Let G be a graph with order n, the matrix A is defined as the adjacency matrix of graph G, which is de-. The number of nonzero eigenvalues and zero eigenvalues in the spectrum of the graph G are called the rant and nullity of the graph G, and denoted by r (G) and η (G) , respectively, obviously r (G) +η (G) = n. Sinogowitz put forward a question that how to find out all singular graphs (equivalently, to show all the nonsingular graphs), namely, to describe all graphs with nullity are greater than zero, it was a very difficult problem only with some special results [3] [4] [5] [6] [7]. By using three graph transformations which do not change the singularity, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained. The notation and terminology that are not described here are provided in [1]

Some Lemmas
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